The present invention relates to methods for generating complex waveforms and modulated signals. Using conventional mathematical techniques, it is difficult to express many output functions commonly used in telecommunications, computing, machine control and other electronic systems. For example, using conventional techniques it is impossible to express a single square pulse having a value of one over a short input range and value of zero elsewhere. At least two equations are required (y=1, y=0) along with restrictions as to the locations where one equation or the other holds sway.
Another problem in mathematically expressing square pulses, binary bit streams and other common waveforms is the mathematical difficulty in expressing discontinuities or singularities such as those that occur at the transitions between 0 and 1 and from 1 to 0 in binary signals. Theoretically these transitions take place instantaneously, the output function being undefined at the point of transition. Practically, this can lead to uncertainty when such functions are applied to electronic circuits and the like. Intermediate values can be recorded at the transitions, leading to ambiguous or inaccurate results.
Such singularities occur in many commonly used waveforms. Step functions, gate pulses, impulse functions, and the like, all raise difficult problems of mathematical expression for scientists and engineers who work with such waveforms everyday. In fact, solutions to these types of problems must be designed into the hardware circuitry of many digital electronics devices. Timing circuits, error correcting circuits and the like are necessary to verify data and synchronize signals to ensure that bits are not being read during binary transitions, for example. This adds to the cost, complexity and size of many devices.
Conventional mathematics for encoding data onto binary and other signals are also complex. Many modulation techniques such as pulse width modulation, time division multiplexing (TDM), code division multiple access, and the like, all require extensive mathematical algorithms to implement. Again, the complexity of the mathematics results in complex hardware and software solutions for generating and using such signals. Synchronization is also a problem with many current digital systems. Ensuring that bits are read at the proper times and other timing and phase shift issues can greatly add to the complexity of digital systems.
Finally, because the binary transitions in digital signals and singularities in other discontinuous functions are undefined, they represent a portion of a signal that may not be used for carrying information. The density of data carried by a binary signal can be greatly increased if it is possible to mathematically define the transitions between ones to zeroes and zeroes to ones in an easy convenient manner and in such a way that the transitions themselves are capable of carrying data.
In light of the above, there presently exists a need for improved methods of generating and expressing complex waveforms. Improved methods of generating waveforms should accommodate waveforms which have traditionally been difficult to express mathematically such as step functions, gate functions and the like. Furthermore, the output position of such functions should be controlled with great precision so that it may be accurately predicted when pulses should or should not occur.
New modulation techniques are also needed for implementing such improved methods of generating complex waveforms. Ideally, such techniques should allow every aspect of a complex waveform to be determined by either a single function or by cascading a series of like functions. Furthermore, such improved techniques should allow for multiple modulation schemes to be added to a single carrier so that the density of data stored on the carrier may be greatly increased.